Tunnel effect

Tunneling effect , tunneling — the microparticle overcomes a potential barrier when its total energy (which remains unchanged during tunneling) is less than the barrier height. The tunnel effect is a phenomenon of exclusively quantum nature, impossible in classical mechanics and even completely contradicting it. An analogue of the tunneling effect in wave optics can be the penetration of a light wave into a reflecting medium (over distances of the order of the light wavelength) under conditions when, from the point of view of geometric optics , total internal reflection occurs. The phenomenon of tunneling underlies many important processes in atomic and molecular physics, in the physics of the atomic nucleus , solid body , etc.

Content

1 Brief quantum mechanical description
2 Potential barrier transparency coefficient
3 Simplified explanation
4 Macroscopic manifestations of the tunnel effect
5 History and researchers
6 See also
7 notes
8 References
9 Literature

Short quantum mechanical description

Reflection and tunneling of an electron beam directed at a potential barrier . The dim spot to the right of the barrier is the electrons passing through the barrier. Pay attention to the interference between incident and reflected waves.

According to classical mechanics, a particle can only be at those points in space at which its potential energy is ${U_{\rm {pot}}}$ ${\ displaystyle {U _ {\ rm {pot}}}}$ - less than full. This follows from the fact that the kinetic energy of a particle

{E_{k}}={\frac {p^{2}}{2m}}={E}-{U_{\rm {pot}}}

{\ displaystyle {E_ {k}} = {\ frac {p ^ {2}} {2m}} = {E} - {U _ {\ rm {pot}}}}

cannot (in classical physics) be negative, since in this case the momentum will be an imaginary value . That is, if two regions of space are separated by a potential barrier, such that ${U_{\rm {pot}}}>{E}$ ${\ displaystyle {U _ {\ rm {pot}}}> {E}}$ , the seepage of a particle through it in the framework of the classical theory is impossible.

In quantum mechanics, the imaginary value of the particle momentum corresponds to the exponential dependence of the wave function on its coordinate. This shows the Schrödinger equation with constant potential (a simplified Schrödinger equation in the one-dimensional case):

{\frac {{{\rm {d}}^{2}}{\psi }}{{{\rm {d}}{x}}^{2}}}+{\frac {2m}{{\hbar }^{2}}}{\left({E}-{U_{\rm {pot}}}\right)}{\psi }=0

{\ displaystyle {\ frac {{{\ rm {d}} ^ {2}} {\ psi}} {{{\ rm {d}} {x}} ^ {2}}} + {\ frac {2m } {{\ hbar} ^ {2}}} {\ left ({E} - {U _ {\ rm {pot}}} \ right)} {\ psi} = 0}

Where $x~-$ ${\ displaystyle x ~ -}$ coordinate ; $E~-$ ${\ displaystyle E ~ -}$ total energy $U_{\rm {pot}}~-$ ${\ displaystyle U _ {\ rm {pot}} ~ -}$ potential energy ${\hbar ~-}$ ${\ displaystyle {\ hbar ~ -}}$ reduced Planck constant , $m~-$ ${\ displaystyle m ~ -}$ particle mass ).

If

E>{U_{\rm {pot}}}

{\ displaystyle E> {U _ {\ rm {pot}}}}

, then the solution to this equation is the function:

{\psi }=A\exp {\left(ix{\frac {\sqrt {2m{\left({E}-{U_{\rm {pot}}}\right)}}}{\hbar }}\right)+B\exp \left(-ix{\frac {\sqrt {2m{\left({E}-{U_{\rm {pot}}}\right)}}}{\hbar }}\right)}

{\ displaystyle {\ psi} = A \ exp {\ left (ix {\ frac {\ sqrt {2m {\ left ({E} - {U _ {\ rm {pot}}} \ right)}}} {\ hbar}} \ right) + B \ exp \ left (-ix {\ frac {\ sqrt {2m {\ left ({E} - {U _ {\ rm {pot}}} \ right)}}} {\ hbar }} \ right)}}

Let there be a moving particle along the path of which there is a potential barrier of height $U_{0}>>E$ ${\ displaystyle U_ {0} >> E}$ , and the particle potential before and after the barrier $U_{f}<E$ ${\ displaystyle U_ {f} <E}$ . Let also the beginning of the barrier coincide with the origin, and its “width” is equal to $a$ ${\ displaystyle a}$ .

For areas $I$ ${\ displaystyle I}$ (before passing) $I I$ ${\ displaystyle II}$ (while passing inside the potential barrier) and $I I I$ ${\ displaystyle III}$ (after passing the barrier), respectively, the functions are obtained:

{{\psi }_{I}}={A_{1}}\exp {\left(ikx\right)}+{B_{1}}\exp {\left(-ikx\right)}

{\ displaystyle {{\ psi} _ {I}} = {A_ {1}} \ exp {\ left (ikx \ right)} + {B_ {1}} \ exp {\ left (-ikx \ right)} }

{{\psi }_{II}}={A_{2}}\exp {\left(-{\chi }x\right)}+{B_{2}}\exp {\left({\chi }x\right)}

{\ displaystyle {{\ psi} _ {II}} = {A_ {2}} \ exp {\ left (- {\ chi} x \ right)} + {B_ {2}} \ exp {\ left ({ \ chi} x \ right)}}

{{\psi }_{III}}={A_{3}}\exp {\left(ik(x-a)\right)}+{B_{3}}\exp {\left(-ik(x-a)\right)}

{\ displaystyle {{\ psi} _ {III}} = {A_ {3}} \ exp {\ left (ik (xa) \ right)} + {B_ {3}} \ exp {\ left (-ik ( xa) \ right)}}

Where $k={\sqrt {{\frac {2m}{\hbar ^{2}}}{\left({E}-{U_{f}}\right)}}}$ ${\ displaystyle k = {\ sqrt {{\ frac {2m} {\ hbar ^ {2}}} {\ left ({E} - {U_ {f}} \ right)}}}}$ , $\chi ={\sqrt {{\frac {2m}{\hbar ^{2}}}{\left({U_{0}-E}\right)}}}$ ${\ displaystyle \ chi = {\ sqrt {{\ frac {2m} {\ hbar ^ {2}}} {\ left ({U_ {0} -E} \ right)}}}}$

Since the term ${B_{3}}\exp {\left(-ik(x-a)\right)}$ ${\ displaystyle {B_ {3}} \ exp {\ left (-ik (xa) \ right)}}$ characterizes the reflected wave coming from infinity, which is absent in this case, it is necessary to put ${B_{3}}=0$ ${\ displaystyle {B_ {3}} = 0}$ . To characterize the magnitude of the tunneling effect, a barrier transparency coefficient is introduced, which is equal to the modulus of the ratio of the flux density of transmitted particles to the flux density of fallen particles:

D={\frac {j_{III}}{j_{I}}}

{\ displaystyle D = {\ frac {j_ {III}} {j_ {I}}}}

The following formula is used to determine particle flux:

{j}={\frac {i{\hbar }}{2m}}{\left({\frac {{\partial }{{\psi }^{*}}}{{\partial }x}}{\psi }-{\frac {{\partial }{\psi }}{{\partial }x}}{{\psi }^{*}}\right)}

{\ displaystyle {j} = {\ frac {i {\ hbar}} {2m}} {\ left ({\ frac {{\ partial} {{\ psi} ^ {*}}} {{\ partial} x }} {\ psi} - {\ frac {{\ partial} {\ psi}} {{\ partial} x}} {{\ psi} ^ {*}} \ right)}}

where the * sign denotes complex conjugation .

Substituting the wave functions indicated above into this formula, we obtain

{D}={\frac {|{A_{3}}|^{2}}{|{A_{1}}|^{2}}}

{\ displaystyle {D} = {\ frac {| {A_ {3}} | ^ {2}} {| {A_ {1}} | ^ {2}}}}

Now, using the boundary conditions, we first express $A_{2}$ ${\ displaystyle A_ {2}}$ and $B_{2}$ ${\ displaystyle B_ {2}}$ across $A_{3}$ ${\ displaystyle A_ {3}}$ (taking into account that ${\chi }a~{\gg }~1$ ${\ displaystyle {\ chi} a ~ {\ gg} ~ 1}$ ):

{A_{2}}={\frac {1-in}{2}}{A_{3}}{\exp {\left({\chi }a\right)}}~,~~~~~~{B_{2}}={\frac {1+in}{2}}{A_{3}}{\exp {\left(-{\chi }a\right)}}~{\approx }~0

{\ displaystyle {A_ {2}} = {\ frac {1-in} {2}} {A_ {3}} {\ exp {\ left ({\ chi} a \ right)}} ~, ~~~ ~~~ {B_ {2}} = {\ frac {1 + in} {2}} {A_ {3}} {\ exp {\ left (- {\ chi} a \ right)}} ~ {\ approx } ~ 0}

n={\frac {k}{\chi }}={\sqrt {\frac {E-U_{f}}{{U_{0}}-E}}}

{\ displaystyle n = {\ frac {k} {\ chi}} = {\ sqrt {\ frac {E-U_ {f}} {{U_ {0}} - E}}}}

and then $A_{1}$ ${\ displaystyle A_ {1}}$ across $A_{3}$ ${\ displaystyle A_ {3}}$ :

{A_{1}}={i{\frac {\left(1-in\right)^{2}}{4n}}}{\exp {\left({\chi }a\right)}}{A_{3}}

{\ displaystyle {A_ {1}} = {i {\ frac {\ left (1-in \ right) ^ {2}} {4n}}} {\ exp {\ left ({\ chi} a \ right) }} {A_ {3}}}

We introduce the quantity

{D_{0}}={\frac {16{n^{2}}}{{\left(1+{n^{2}}\right)}^{2}}}=16{\frac {(U_{0}-E)(E-U_{f})}{(U_{0}-U_{f})^{2}}}

{\ displaystyle {D_ {0}} = {\ frac {16 {n ^ {2}}} {{\ left (1+ {n ^ {2}} \ right)} ^ {2}}} = 16 { \ frac {(U_ {0} -E) (E-U_ {f})} {(U_ {0} -U_ {f}) ^ {2}}}}

which will be of the order of unity. Then:

D~{\cong }~{D_{0}}{\exp {\left(-{\frac {2a{\sqrt {2m{\left({U_{0}}-E\right)}}}}{\hbar }}\right)}}

{\ displaystyle D ~ {\ cong} ~ {D_ {0}} {\ exp {\ left (- {\ frac {2a {\ sqrt {2m {\ left ({U_ {0}} - E \ right)} }}} {\ hbar}} \ right)}}}

For a potential barrier of arbitrary shape, we make a replacement

{\frac {2a{\sqrt {2m{\left({U_{0}}-E\right)}}}}{\hbar }}~{\Rrightarrow }~{\frac {2}{\hbar }}{\int \limits _{x_{1}}^{x_{2}}{\sqrt {2m{\left({U(x)}-E\right)}}}\,{\rm {d}}x}

{\ displaystyle {\ frac {2a {\ sqrt {2m {\ left ({U_ {0}} - E \ right)}}}} {\ hbar}} ~ {\ Rrightarrow} ~ {\ frac {2} { \ hbar}} {\ int \ limits _ {x_ {1}} ^ {x_ {2}} {\ sqrt {2m {\ left ({U (x)} - E \ right)}}} \, {\ rm {d}} x}}

Where $x_{1}$ ${\ displaystyle x_ {1}}$ and $x_{2}$ ${\ displaystyle x_ {2}}$ are out of condition

{U(x_{1})}={U(x_{2})}=E

{\ displaystyle {U (x_ {1})} = {U (x_ {2})} = E}

Then for the coefficient of passage through the barrier we obtain the expression

D~{\cong }~{D_{0}}{\exp {\left(-{\frac {2}{\hbar }}{\int \limits _{x_{1}}^{x_{2}}{\sqrt {2m{\left({U(x)}-E\right)}}}\,{\rm {d}}x}\right)}}

{\ displaystyle D ~ {\ cong} ~ {D_ {0}} {\ exp {\ left (- {\ frac {2} {\ hbar}} {\ int \ limits _ {x_ {1}} ^ {x_ {2}} {\ sqrt {2m {\ left ({U (x)} - E \ right)}}}, {\ rm {d}} x} \ right)}}}

Potential Barrier Transparency

The transparency coefficient of a potential barrier is the ratio of the flux density of particles passing through the barrier to the flux density of particles incident on the barrier. For a barrier of arbitrary shape, it is approximately equal to:

D~{\cong }~C{\exp {\left(-{\frac {2}{\hbar }}{\int \limits _{x_{1}}^{x_{2}}{\sqrt {2m{\left({U(x)}-E\right)}}}\,{\rm {d}}x}\right)}}

{\ displaystyle D ~ {\ cong} ~ C {\ exp {\ left (- {\ frac {2} {\ hbar}} {\ int \ limits _ {x_ {1}} ^ {x_ {2}} { \ sqrt {2m {\ left ({U (x)} - E \ right)}}} \, {\ rm {d}} x} \ right)}}}

Where $C$ ${\ displaystyle C}$ - coefficient of the order of 1, $x_{1},x_{2}$ ${\ displaystyle x_ {1}, x_ {2}}$ - coordinates of points for which $U(x)=E$ ${\ displaystyle U (x) = E}$ , $x_{2}-x_{1}$ ${\ displaystyle x_ {2} -x_ {1}}$ - width of the barrier for a particle with energy $E$ ${\ displaystyle E}$ , $U_{max}-E$ ${\ displaystyle U_ {max} -E}$ - the height of the barrier ^[1] .

Simplified explanation

The tunnel effect can be explained by the uncertainty relation . ^[2] Recorded as:

\Delta x\Delta p\geqslant {\frac {\hbar }{2}}

{\ displaystyle \ Delta x \ Delta p \ geqslant {\ frac {\ hbar} {2}}}

,

it shows that when a quantum particle is limited in coordinate, that is, an increase in its definiteness in x , its momentum p becomes less defined. Randomly impulse uncertainty $\Delta p$ ${\ displaystyle \ Delta p}$ can add a particle of energy to overcome the barrier. Thus, with some probability, a quantum particle can penetrate the barrier, - this probability is greater, the smaller the mass of the particle, the narrower the potential barrier and the less energy the particle lacks to reach the height of the barrier, the average energy of the penetrated particle will remain unchanged.

It follows from the formula for the coefficient of passage through a barrier that particles pass through a potential barrier in a noticeable way only at its thickness $l$ ${\ displaystyle l}$ defined by approximate equality ${\frac {2}{\hbar }}{\sqrt {2m{\left(U_{m}-E\right)}}}l\approx 1$ ${\ displaystyle {\ frac {2} {\ hbar}} {\ sqrt {2m {\ left (U_ {m} -E \ right)}}} l \ approx 1}$ . Here $U_{m}$ ${\ displaystyle U_ {m}}$ - maximum barrier height. To detect a particle inside a potential barrier, we must measure its coordinate with an accuracy not exceeding the depth of its penetration $\Delta x<l$ ${\ displaystyle \ Delta x <l}$ . It follows from the uncertainty principle that in this case the particle momentum acquires ${\bar {\Delta p^{2}}}>{\frac {\hbar ^{2}}{4{\bar {\Delta x^{2}}}}}={\frac {\hbar ^{2}}{4l^{2}}}$ ${\ displaystyle {\ bar {\ Delta p ^ {2}}}> {\ frac {\ hbar ^ {2}} {4 {\ bar {\ Delta x ^ {2}}}} = {\ frac { \ hbar ^ {2}} {4l ^ {2}}}}$ . Value $l$ ${\ displaystyle l}$ can be found from the formula ${\frac {2}{\hbar }}{\sqrt {2m{\left(U_{m}-E\right)}}}l\approx 1$ ${\ displaystyle {\ frac {2} {\ hbar}} {\ sqrt {2m {\ left (U_ {m} -E \ right)}}} l \ approx 1}$ , as a result we get ${\frac {\bar {\Delta p^{2}}}{2m}}>U_{m}-E$ ${\ displaystyle {\ frac {\ bar {\ Delta p ^ {2}}} {2m}}> U_ {m} -E}$ .

Thus, the kinetic energy of a particle when passing through a barrier increases by the amount required to pass through the barrier as a result of the uncertainty of its momentum, which is determined by the uncertainty principle as a result of the uncertainty of measuring its coordinate ^[3] .

Macroscopic manifestations of the tunnel effect

Tunnel diode and jumper .

The tunneling effect has a number of manifestations in macroscopic systems:

Tunneling of charge carriers through the potential barrier of the pn junction , which has gained practical application in a tunnel diode .
Tunneling of charge carriers through a thin oxide film having dielectric properties, covering a number of metals (in particular, aluminum ) and ensuring the conductivity of the points of mechanical connection of conductors (twisted wires, clamps, jumpers ). As applied to superconductors, this phenomenon is called the Josephson effect .

History and Researchers

The discovery of the tunneling effect was preceded by the discovery of radioactive decay by A. Beckerel in 1896, the study of which was continued by the spouses Maria and Pierre Curie , who in 1903 received the Nobel Prize for their research ^[4] . Based on their research in the next decade, the theory of radioactive half-life was formulated, which was soon confirmed experimentally.

At the same time, in 1901, a young scientist Robert Francis Earhart, who studied the behavior of gases between electrodes in various modes using an interferometer , unexpectedly received inexplicable data. After reviewing the results of the experiments, the famous scientist D. Thomson suggested that the law that has not yet been described is in effect and called on scientists to further research. In 1911 and 1914, one of his graduate students , Franz Rother, repeated Earhart's experiment, using a more sensitive galvanometer instead of an interferometer for measurements, and definitely fixed an unexplained stationary field of electron emission between the electrodes. In 1926, the same Roser used in experiment the latest galvanometer with a sensitivity of 26 pA and fixed a stationary field of electron emission arising between closely spaced electrodes even in deep vacuum ^[5] .

In 1927, the German physicist Friedrich Hund was the first to mathematically reveal the “tunnel effect” in calculating the rest of the double-well potential ^[4] . In 1928, independently of each other, the tunnel effect formulas were used in their works by the Russian scientist George Gamow and the American scientists and Edward Conndon in developing the theory of alpha decay ^[6] ^[7] ^[8] ^[9] ^[10] . Both studies simultaneously solved the Schrödinger equation for the nuclear potential model and mathematically substantiated the relationship between the radioactive half-life of particles and their radioactive emission, the tunneling probability.

Having attended the Gamow seminar, the German scientist Max Born successfully developed his theory, suggesting that the “tunneling effect” is not limited to the field of nuclear physics, but has a much broader effect, since it arises according to the laws of quantum mechanics and, therefore, is applicable to describe phenomena in many other systems ^[11] . In case of autonomous emission from metal into vacuum, for example, according to the “Fowler – Nordheim law” formulated in the same 1928.

In 1957, the study of semiconductors , the development of transistor and diode technologies, led to the discovery of electron tunneling in mechanical particles. In 1973, American David Josephson received the Nobel Prize in Physics “For Theoretical Prediction of the Properties of the Superconductivity Passing Through a Tunnel Barrier,” along with him, Japanese Leo Esaki and Norwegian Ivar Giever “For experimental discoveries of tunneling phenomena in semiconductors and superconductors respectively” ^{[[ 11]} In 2016, the " " ^[12] was also discovered.

Notes

↑ Yavorsky B.M. , Detlaf A.A. , Lebedev A.K. Physics handbook for engineers and university students. - M .: Onyx , 2007 .-- ISBN 978-5-488-01248-6 . - Circulation 5 100 copies. - S. 774
↑ Article “Tunnel Effect” in TSB , 2 paragraph
↑ Blokhintsev D.I. Fundamentals of quantum mechanics. - M., Higher School, 1961. - c. 330
↑ ¹ ² Nimtz. Zero Time Space / Nimtz, Haibel. - Wiley-VCH, 2008 .-- P. 1.
↑ Thomas Cuff. The STM (Scanning Tunneling Microscope) [The forgotten contribution of Robert Francis Earhart to the discovery of quantum tunneling. ] (unspecified) . ResearchGate .
↑ G. Gamow. Essay on the development of the doctrine of the structure of the atomic nucleus (I. Theory of radioactive decay) // UFN 1930. V. 4.
↑ Gurney, RW; Condon, EU Quantum Mechanics and Radioactive Disintegration (Eng.) // Nature. - 1928. - Vol. 122 , no. 3073 . - P. 439 . - DOI : 10.1038 / 122439a0 . - .
↑ Gurney, RW; Condon, EU Quantum Mechanics and Radioactive Disintegration (Neopr.) // Phys. Rev. - 1929. - T. 33 , No. 2 . - S. 127-140 . - DOI : 10.1103 / PhysRev . 33.127 . - .
↑ Bethe, Hans (October 27, 1966), Hans Bethe - Session I. Interview with Charles Weiner; Jagdish Mehra , Cornell University, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA , < https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4504- 1 > . Retrieved May 1, 2016.
↑ Friedlander, Gerhart. Nuclear and Radiochemistry / Gerhart Friedlander, Joseph E. Kennedy, Julian Malcolm Miller. - 2nd. - New York: John Wiley & Sons, 1964. - P. 225–7. - ISBN 978-0-471-86255-0 .
↑ ¹ ² Razavy, Mohsen. Quantum Theory of Tunneling. - World Scientific, 2003. - P. 4, 462. - ISBN 9812564888 .
↑ Quantum Tunneling of Water in Beryl: A New State of the Water Molecule (neopr.) . Physical Review Letters (April 22, 2016). doi : 10.1103 / PhysRevLett.116.167802 . Date of treatment April 23, 2016.

Links

Tunnel emission - an article from the Great Soviet Encyclopedia .
Tunneling effect // Big Soviet Encyclopedia : [in 30 t.] / Ch. ed. A.M. Prokhorov . - 3rd ed. - M .: Soviet Encyclopedia, 1969-1978.

Literature

Goldansky V.I., Trakhtenberg L.I., Flerov V.N. Tunneling phenomena in chemical physics. M .: Nauka, 1986 .-- 296 p.
Blokhintsev D.I., Fundamentals of quantum mechanics, 4th ed., M., 1963;
Landau, L.D. , Lifshits, E.M. Quantum mechanics (nonrelativistic theory). - 3rd edition, revised and supplemented. - M .: Nauka , 1974.- 752 p. - (“ Theoretical Physics ”, Volume III).

[Javor-1] Yavorsky B.M. , Detlaf A.A. , Lebedev A.K. Physics handbook for engineers and university students. - M .: Onyx , 2007 .-- ISBN 978-5-488-01248-6 . - Circulation 5 100 copies. - S. 774

[2] Article “Tunnel Effect” in TSB , 2 paragraph

[3] Blokhintsev D.I. Fundamentals of quantum mechanics. - M., Higher School, 1961. - c. 330

[Nimtz-4] ¹ ² Nimtz. Zero Time Space / Nimtz, Haibel. - Wiley-VCH, 2008 .-- P. 1.

[5] Thomas Cuff. The STM (Scanning Tunneling Microscope) [The forgotten contribution of Robert Francis Earhart to the discovery of quantum tunneling. ] (unspecified) . ResearchGate .

[6] G. Gamow. Essay on the development of the doctrine of the structure of the atomic nucleus (I. Theory of radioactive decay) // UFN 1930. V. 4.

[7] Gurney, RW; Condon, EU Quantum Mechanics and Radioactive Disintegration (Eng.) // Nature. - 1928. - Vol. 122 , no. 3073 . - P. 439 . - DOI : 10.1038 / 122439a0 . - .

[8] Gurney, RW; Condon, EU Quantum Mechanics and Radioactive Disintegration (Neopr.) // Phys. Rev. - 1929. - T. 33 , No. 2 . - S. 127-140 . - DOI : 10.1103 / PhysRev . 33.127 . - .

[9] Bethe, Hans (October 27, 1966), Hans Bethe - Session I. Interview with Charles Weiner; Jagdish Mehra , Cornell University, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA , < https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4504- 1 > . Retrieved May 1, 2016.

[Nuc&RadChem-10] Friedlander, Gerhart. Nuclear and Radiochemistry / Gerhart Friedlander, Joseph E. Kennedy, Julian Malcolm Miller. - 2nd. - New York: John Wiley & Sons, 1964. - P. 225–7. - ISBN 978-0-471-86255-0 .

[разави-11] ¹ ² Razavy, Mohsen. Quantum Theory of Tunneling. - World Scientific, 2003. - P. 4, 462. - ISBN 9812564888 .

[12] Quantum Tunneling of Water in Beryl: A New State of the Water Molecule (neopr.) . Physical Review Letters (April 22, 2016). doi : 10.1103 / PhysRevLett.116.167802 . Date of treatment April 23, 2016.